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  Literature question: Generalized Dirac equation

+ 5 like - 0 dislike

For the Dirac equation, $i\gamma^\mu\partial_\mu\psi-m\psi=0$, $\overline{\psi}\gamma^\mu\psi$ is a conserved current. I feel like I've known this since I was three. $\overline{\psi}\gamma^\mu\psi$ is also a conserved constant, however, for the related parameterized set of equations,


provided $\alpha_1$, $\alpha_2$, and $\alpha_3$ are real constants. A further constraint in QFT is that the matrix

$$M=k_\alpha\gamma^\alpha\left[k_\mu\gamma^\mu(1+i\alpha_1\gamma^5)-m(\alpha_3+i\alpha_2\gamma^5)\right],\quad \mbox{where } k_\mu k^\mu=m^2,$$

must be positive semi-definite for the two-point VEVs to be positive semi-definite, as they must be for us to construct a free field Fock-Hilbert space, which is satisfied only if $\alpha_1^2+\alpha_2^2+\alpha_3^2\le 1$. Given that, we have a class of free quantum fields. If $\alpha_1^2+\alpha_2^2+\alpha_3^2=1$, $M$ has a 2-dimensional zero eigenspace, as for the usual Dirac equation, $\alpha_3=1,\alpha_2=\alpha_1=0$, or in it's conjugate form, $\alpha_3=-1,\alpha_2=\alpha_1=0$, so achieving the bound is perhaps preferred so as not to introduce too many DoFs.

Is this generalized Dirac equation discussed in the literature? It seems possible that electrons, muons, and tauons might satisfy this equation with different values of $\alpha_1$, $\alpha_2$, and $\alpha_3$, but yet with the same conserved current, and that this difference might make a difference, or at least that someone must have shown that this either isn't useful or is equivalent to the usual Dirac equation. We also might investigate symmetries that transform between different values of $\alpha_1$, $\alpha_2$, and $\alpha_3$, etc., etc.

References preferred, or else an explanation of why it's obvious why this isn't useful. Thanks.

asked Jan 18, 2016 in Resources and References by Peter Morgan (1,230 points) [ no revision ]
recategorized Jan 18, 2016 by Dilaton

I haven't seen this equation before. Possibly because it might be the case that under the constraints given there is a linear field transformation that transforms the equation into the standard Dirac equation. Did you try this?

@ArnoldNeumaier There is an SU(2) action on vectors of the form


generated by $\gamma^5C$, $\frac{k_\mu\gamma^\mu}{m}C$, and $\frac{k_\mu\gamma^\mu\gamma^5}{m}$, where $C$ is a charge conjugation operator that commutes with $\gamma^\mu$, with $C^2=1$ and $CiC=-i$. That expressing this symmetry requires the introduction of $C$ suggests why this generalized Dirac equation is not mentioned as a possibility, but in any case $\overline{\psi}\gamma^\mu\psi$ is conserved; $C$ only has to be invoked to transform between the different cases. The specialness of $C$ could be a spontaneous breaking of the symmetry.

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